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Math Help - Isomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    Isomorphism

    Define for each \lambda\in\mathbb{R} a natural action on T_{\alpha} by rotation:
    <br />
 \tau(\lambda)(f)(t) = \left\{<br />
     \begin{array}{lr}<br />
       f(t+\lambda) & : t+\lambda \in [0,1]\\<br />
       \alpha^n(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z}<br />
     \end{array}<br />
   \right.<br />
    where \alpha is an automorphism of the C^*-algebra A. Prove that the one parameter family \lambda\mapsto \tau(\lambda) gives rise to an isomorphism

    <br />
 T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))<br />

    where T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}
    Last edited by Mauritzvdworm; September 27th 2010 at 11:42 AM. Reason: typing error
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    Define for each \lambda\in\mathbb{R} a natural action on T_{\alpha} by rotation:
    <br />
 \tau(\lambda)(f)(t) = \left\{<br />
     \begin{array}{lr}<br />
       f(t+\lambda) & : t+\lambda \in [0,1]\\<br />
       \alpha(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z}<br />
     \end{array}<br />
   \right.<br />
    where \alpha is an automorphism of the C^*-algebra A. Prove that the one parameter family \lambda\mapsto \tau(\lambda) gives rise to an isomorphism

    <br />
 T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))<br />

    where T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}
    I can't help with this problem, which probably requires a fairly extensive knowledge of semidirect products (and in any case I am about to go away for a week). But the problem as stated has two obvious mistakes.

    First, the definition of \tau(\lambda) is wrong. It does not define an automorphism group. In fact, the definition of \tau(\lambda)(f)(t) does not even define a continuous function of t at integer values of t+\lambda. The correct definition should presumably be \tau(\lambda)(f)(t) = \alpha^n(f(t+\lambda-n)) when t+\lambda\in[n,n+1].

    Second, the semidirect product T_\alpha\rtimes_\alpha\mathbb{R} does not make sense. It should be T_\alpha\rtimes_\lambda\mathbb{R}.
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